JERUSALEM—Israel is careening into a new crisis on the heels of the Gaza war—an escalation of violence in East Jerusalem unseen in more than a decade that could pit U.S. allies against each other over Jewish settlements and holy sites.

On Thursday, Israel completely shut one of Islam’s holiest sites in the Old City for the first time in 14 years. The last time authorities barred all entry to the site known to Jews as the Temple Mount...